On the Viability of Gravitational Bose-Einstein Condensates As

Draft version November 15, 2018 Preprint typeset using LATEX style emulateapj v. 11/10/09 ON THE VIABILITY OF GRAVITATIONAL BOSE-EINSTEIN CONDENSATES AS ALTERNATIVES TO SUPERMASSIVE BLACK HOLES Hujeirat, A.A.1 Draft version November 15, 2018 ABSTRACT Black holes are inevitable mathematical outcome of spacetime-energy coupling in general relativity. Currently these objects are of vital importance for understanding numerous phenomena in astrophysics and cosmology. However, neither theory nor observations have been capable of unequivocally prove the existence of black holes or granting us an insight of what their internal structures could look like, therefore leaving researchers to speculate about their nature. In this paper the reliability of supermassive Bose-Einstein condensates (henceforth SMBECs) as alternative to supermassive black holes is examined. Such condensates are found to suffer of a causality problem that terminates their cosmological growth toward acquiring masses typical for quasars and triggers their self-collapse into supermassive black holes (SMBHs). It is argued that a SMBEC-core most likely would be subject to an extensive deceleration of its rotational frequency as well as to vortex-dissipation induced by the magnetic fields that thread the crust, hence diminishing the superfluidity of the core. Given that rotating superfluids between two concentric spheres have been verified to be dynamically unstable to non axi-symmetric perturbations, we conclude that the remnant energy stored in the core would be sufficiently large to turn the flow turbulent and dissipative and subsequently lead to core collapse bosonova. The feasibility of a conversion mechanism of normal matter into bosonic condensates under normal astrophysical conditions is discussed as well. We finally conclude that in lack of a profound theory for quantum gravity, BHs should continue to be the favorite proposal for BH candidates. Subject headings: Relativity: general, neutron stars, black hole, dark objects, boson objects — con- densed matter physics: superfluids, Bose-Einstein condensate — cosmology: dark energy — fluids: superfluids, superconduction 1. INTRODUCTION Obviously, unless there is a unusually strong accumu- Before 227 years ago and 118 after Newton’s gravi- lation of energy in a relatively small volume, the LHS tation (1666), John Michell (1787) was the first to an- and the RHS of Eq. (1) can be conveniently decoupled, therefore justifying the Newtonian theory in the weak ticipate the existence of a critical radius, RH , below which even the light cannot escape the gravitational pull field limit in flat spacetime. of the central point mass object. Based on Newtonian Shortly thereafter the GR was published in (1915), arXiv:1109.3821v2 [gr-qc] 20 Sep 2011 gravity, he equated the potential energy E of the ob- Karl Schwarzschild presented a solution for the field p equations that corresponds to a spherically symmetric ject of mass M to the kinetic energy EK and obtained: 2 object of mass M embedded in vacuum. This solution Rcrit =2GM/c , where ”c, G” are the speed of light and the gravitational constant, though the speed of light was was the first theoretical proof that BHs are inevitable still uncertain and the possible existence of such light- products of general relativity (GR), where the spacetime capturing objects in nature was merely a fictitious pro- becomes indefinitely warped. posal. In 1915 presented Einstein his profound theory of grav- Several years later, Subrahmanyan Chandrasekar itation, therein postulating the energy as a field in four- (1931) proposed a framework for the BH-creation. dimensional spacetime and posting it as source for cur- Accordingly, sufficiently massive stars could in principle vature. Mathematically, Einstein field equations read: collapse under their own self-gravity, as neither the thermal nor the degenerate pressures could counter-balance Gµν = κTµν , (1) the enormous gravitational attraction. where Gµν , Tµν are the 2-rank Einsteins’s geometrical From the point of view of Schwarzschild’s solution, a and energy-momentum tensors, respectively. The coeffi- massive star could contract in a quasi-stationary manner cient κ =8πG/c2 1.863 10−27. to finally undergo a dynamical collapse into indefinitely ≈ × small size at the center. However, a sufficiently distant 1 IWR - Interdisciplinary Center for Scientific Computing, observer would be able to observe the collapsing matter University of Heidelberg, INF 368, 69120 Heidelberg, Germany up to the horizon, but not beyond. 2 In the early times, this scenario was rejected both by theorists and observers alike and in particular by Einstein and Eddington. Only at the beginning of the sixties the BH-proposal was revived, when BHs appeared to be the only reasonable objects to explain the origin of the vast energy output observed to liberate from quasars. Since then the BH-proposal has been repeatedly adopted and their mass-regime has been continuously expanded to finally span the whole mass spectrum, ranging from the microgram scale up to billion solar Figure 1. The gravitational field of a non-rotating object masses. of mass M embedded in vacuum is equivalent to a curved spacetime having the event horizon as a boundary. The hy- Although BHs are unobservables by definition, the perspace on the right is a projection of the four-dimensional dynamical behavior of matter and stars in their vicinity curved spacetime. in principle should disclose the properties of these objects. In particular, the distinguished depth of their As it will be shown below, the Schwartzschild’s pro- gravitational well, enables BHs to convert a significant posal raises various fundamental questions in GR rather portion of the potential energy of matter or objects than providing a solution to a problem: approaching the BH into other forms of energy, such as The Schwarzschild solution contains two singular- thermal or magnetic energy. In fact there are additional • ities: r=0 and r = r . While the former is in- observational techniques that are used for predicting s trinsic and unremovable, the latter one can be re- the depth of the gravitational well of BHs, e.g., the iron moved by appropriate coordinate transformation, K α emission lines associated with rotating plasmas in e.g. using the Eddington-Finkelstein coordinates accretion disks, the Lorentz factor of ejected plasmas (Hobson et al. 2006). The contraction of the from their vicinity and possibly through gravitational Riemann curvature tensor at the event horizon: waves detectors to be employed in the near future (see R Rµνστ yields a well-defined finite number, Mueller 2007, for additional detection methods). µνστ which implies that the curvature at this radius is well-behaved and that a co-moving observer will Most BHs in binaries are considered to have masses of feel nothing special as she/he crosses the even hori- the order of 10M⊙. The massive ones are generally found zon. at the center of galaxies with masses ranges between sev- In fact it was shown by Penrose (1965) and eral million up to several billions solar masses. In most Hawking (1967) that if Einstein general theory of cases they are observed to be radiatively active, implying relativity is correct and the stress-energy tensor therefore that they are accreting from or ejecting matter satisfies certain positive-definite inqualties, then into their surrounding. spacetime singularities are inevitable. This may Nevertheless, these techniques hint to the existence of imply that all quantum information generally as- objects that differ from normal or compact stars, such sociated with the matter-building the BH will be as old stars, white dwarfs and neutron stars; however completely destroyed. they definitely are unable to determine their nature. Moreover, as the lightcones tend to close up near the event horizon, r = rs, our connection to Difficulties with the classical BH-Proposal observers approaching the event horizon will be continuously weaker and will be completely lost Consider an object of Mass M and spin=0 embed- inside r = rs, which implies that we will never ded in vacuum (see Fig.1), how does the spacetime know if this observer ever crossed the horizon. around this object look like? From the point of view of GR, the equations to be solved This raises a serious question about the progeni- are Gµν =0. tor of BHs. If these were massive stars that run The solution to this problem is a metric, which was out of energy generation via nuclear fusion at their forwarded to Einstein by Schwarzschild just two months centers followed by a dynamical self-collapse under after the former published his theory of GR. The metric their own self-gravity, then from our point of view reads: as distant observers they must be still collapsing and would cross the event horizon with the speed dr2 ds2 =c2(1 r /r)dt2 dΩ¯ 2, (2) of light after infinite time. Equivalently, the pro- − s − 1 r /r − − s genitors of BHs are massive stars that collapsed into rings of matter that surround the event hori- where ds, c, t, rs correspond to the proper distance zon and whose particles approaching the horizon, between two events in this spacetime, the speed of light, but which will never reach or cross it. the time measured by a fixed observer at infinity and 2 the Schwarzschild radius rs = 2GM/c , respectively. A photon of energy E0 = hν0 measured in our The term dΩ¯ 2 =r2dΩ2 =r2(dθ + sin2θ dϕ2) correspond • frame will be infinitely blue-shifted at the event to a differential area on the surface of a sphere of radius r. horizon. Assuming global energy conservation, this indefinite gain of energy must be extracted from 3 the field and therefore would cause a non-negligible perturbation to the spacetime. As a consequence, each photon that approaches the horizon will cause a measurable change of spacetime curvature around the BH, which mathematically means, that the solution may not be stable against external perturbations.

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